At equilibrium, the state process of player \(i\)  is given by the SDE
\(dX^{i,\underline{\hat{\alpha }}}_t = b(t,X^{i,\underline{\hat{\alpha }}}_t, \hat{\alpha }^{i,N}_t, L^N(\underline{X}_t^{\underline{\hat{\alpha }}}, \hat{\underline{\alpha }}_t))\,dt + \sigma \,dW^i_t\) 
with \(\hat{\alpha }^{i,N}\)  given by (REF ).
Since \(\Lambda \)  is Lipschitz–continuous (this follows from (REF )), it can be shown that \(\varphi _t\)  defined in (REF ) is Lipschitz continuous with respect to the second order Wasserstein distance, see for instance (the proof of) [1]}.
Thus, the functions \(B,F\)  and \(G\)  introduced in REF  are Lipschitz–continuous and of linear growth.
Further observe that using these functions, the McKean–Vlasov equation (REF ) characterizing the mean field equilibrium reads
\({\left\lbrace \begin{array}{ll}d X_t = B(t, X_t, Y_t, \mathcal {L}(X_t, Y_t))\,dt + \sigma \,dW_t\\dY_t = - F(t, X_t, {Y}_t, \mathcal {L}( X_t, Y_t))\,dt + Z_t\,dW_t\\X_0 =x, \quad Y_T = G(X_T, \mathcal {L}(X_T)).\end{array}\right.}\) 
We will now introduce two auxiliary interacting particle systems that will allow us to derive the LDP for the sequence of interest.
First consider the equation
\({\left\lbrace \begin{array}{ll}d\widetilde{X}^{i,N}_t = B(t,\widetilde{X}^{i,N}_t, \widetilde{Y}^{i,N}_t, L^N(\widetilde{\underline{X}}_t, \widetilde{\underline{Y}}_t))\,dt + \sigma \,dW_t^i\\d\widetilde{Y}^{i,N}_t = - F(t,\widetilde{X}^{i,N}_t, \widetilde{Y}^{i,N}_t, L^N(\widetilde{\underline{X}}_t, \widetilde{\underline{Y}}_t))\,dt + \sum _{k=1}^N\widetilde{Z}^{i,k}_t\,dW_t^k\\X_0 = x, \quad \widetilde{Y}_T^{i,N} = G(\widetilde{X}^{i,N}_T,L^N(\underline{\widetilde{X}}_T))\end{array}\right.}\) 
which simply corresponds to (REF ) after taking "\(\varepsilon ^{i,N} = \gamma ^{i,N}= \zeta ^{i,N}=0\) " for all \(i\) .
Note that by Lipschitz-continuity of \(B,F\)  and \(G\) , it follows e.g. from [2]} that if \(T\)  is small enough, then Equation (REF ) admits a unique solution.
The second auxilliary equation is introduced through a decoupling argument.
Since \(V\)  is a classical solution of the PDE (REF ), applying Itô's formula to \(\bar{Y}_t := V(t, X_t, \mathcal {L}(X_t))\)  shows that there is \(\bar{Z}\)  such that \((\bar{Y}, \bar{Z})\)  solves the backward equation in (REF ).
Thus, by uniqueness, we have \(Y_t = V(t, X_t, \mathcal {L}(X_t))\) .
The function \(V\)  is often called a decoupling field for the system (REF ) because it allows to write the system as two decoupled equations, where \(X\)  satisfies
\(dX_t &= B\Big (t,X_t, V(t, X_t, \mathcal {L}(X_t)), \mathcal {L}\big (X_t, V(t, X_t, \mathcal {L}(X_t)))\big ) \Big )\,dt + \sigma \,dW_t\\&= \overline{B}(t,X_t, \mathcal {L}(X_t))\,dt + \sigma \,dW_t\) 

where \(\overline{B}\)  is the function defined in (REF ).
Since \(V(t, \cdot ,\cdot )\)  is Lipschitz–continuous on \(\mathbb {R}^\ell \times \mathcal {P}_2(\mathbb {R}^\ell )\)  uniformly in \(t\in [0,T]\) , it follows that \(\overline{B}\)  is Lipschitz continuous on \(\mathbb {R}^\ell \times \mathcal {P}_2(\mathbb {R}^\ell )\)  as well.
The second auxiliary particle system is then
\(d\mathcal {X}^{i,N}_t = \overline{B}(t,\mathcal {X}^{i,N}_t, L^N(\underline{\mathcal {X}}_t))\,dt + \sigma \,dW^i_t, \quad \mathcal {X}^{i,N}_0 =x.\) 
Note that \((\mathcal {X}^{i,N})_{i=1,\dots ,N}\)  is well-defined by classical SDE theory.
Moreover, the standard theory of propagation of chaos (see e.g. [3]}) shows that the sequence \(\mathcal {X}^{i,N}\)  converges in \(\mathcal {S}^2(\mathbb {R}^\ell )\)  to \(X\)  and the sequence of empirical measures \(L^N(\underline{\mathcal {X}}_t)\)  converges to \(\mathcal {L}(X_t)\)  in \(\mathcal {P}_2(\mathbb {R}^\ell )\) , see e.g. [4]}.
Furthermore, again by Lipschitz–continuity of \(\overline{B}\) , it follows by [5]} (see also [6]}) that the family of empirical measures \((L^N(\underline{\mathcal {X}}))_N\)  satisfies the Laplace principle (in \(\mathcal {P}(\mathcal {C}^\ell )\) ) with rate function given by (REF ).
[5]} shows that this rate function is lower semicontinuous for the weak topology, and therefore the sublevel sets are weakly closed.
Let us show that the rate function is good, i.e. that it has weakly compact sublevel sets.
By Prokhorov's theorem, if we show that the sublevel sets are tight it will follow that they are weakly relatively compact.
Let \(\mathbb {Q}^n\)  be a probability measures on (the Polish space) \(\mathcal {C}^d\)  such that \(\mathcal {I}(\mathbb {Q}^n)\le K\)  for some constant \(K>0\) .
Then there is a sequence \((u^n)\)  in \(\mathcal {H}^2(\mathbb {R}^d)\) , the space of \(\mathbb {R}^d\) –valued, square integrable and progressive processes such that
\(\mathbb {E}\bigg [\frac{1}{2}\int _0^T|u^n_t|^2\,dt\bigg ]< K + 1/n\quad \text{and}\quad \mathcal {L}(X^{u^n}) = \mathbb {Q}^n.\) 
Thus, it suffices to show that the sequence \(\mathcal {L}(X^{u^n})\)  is tight.
Observe that by boundedness of the sequence \((u^n)\)  in \(\mathcal {H}^2(\mathbb {R}^d)\)  and linear growth of \(\overline{B}\)  it follows by standard SDE estimations that \(\sup _n\sup _t\mathbb {E}[|X^{u^n}_t|^2]<\infty \) .
Furthermore, for every \(0\le s\le t\le T\) , we have
\(|X^{u^n}_t - X^{u^n}_s| \le C\int _s^t1 + |X^{u^n}_r| + E[|X^{u^n}_r|^2]^{1/2} + |u^n_r|\,dr + |\sigma ||W_t - W_s|.\) 

Hence, taking expectation on both sides and applying Cauchy-Schwarz inequality yields
\(\mathbb {E}|X^{u^n}_t - X^{u^n}_s| &\le C|t-s|^{1/2}\bigg (1 + \mathbb {E}\bigg [\int _0^T|u^n_r|^2\,dr\bigg ]^{1/2} + \sup _r\mathbb {E}\big [|X^{u^n}_r|^2\big ]^{1/2} + |\sigma |\bigg )\\&\le C|t-s|^{1/2}.\) 

Thus, since \((X^{u^n})\)  is a sequence of continuous processes, it follows by Kolmogorov's tightness criterion that \(\mathbb {Q}^n\)  is tight.
Therefore, \(\mathcal {I}\)  is a good rate function.
In order to "transfer" the LDP from the sequence \((L^N(\underline{\mathcal {X}}))\)  to the relevant sequence \((L^N(\underline{X}^{\underline{\hat{\alpha }}}))\) , we need to show that the two sequences are exponentially close in the sense of Definition REF .
This follows from Chebyshev's inequality and Lemma REF  below since we have
\(\mathbb {P}\Big (\sup _{t\in [0,T]}\mathcal {W}_2(L^N(\underline{X}_t^{\underline{\hat{\alpha }}}), L^N(\underline{\mathcal {X}}_t)) >\varepsilon \Big ) &\le \mathbb {P}\Big ( \Big \lbrace \frac{1}{N}\sup _{t\in [0,T]}\sum _{i=1}^N|X^{\underline{\alpha },i}_t - \mathcal {X}^{i}_t|^2\Big \rbrace ^{1/2} \ge \varepsilon \Big )\\&\le \mathbb {E}\Big [\exp \Big \lbrace  \Big (\sum _{i=1}^N\Vert X^{\underline{\alpha },i} - \mathcal {X}^{i}\Vert ^2_\infty \Big )\Big \rbrace  \Big ]e^{-\varepsilon ^2N^2}\\&\le C e^{-\varepsilon ^2 N^2}.\) 
Therefore,
\(\lim _{N\rightarrow \infty }\frac{1}{N}\log \mathbb {P}\Big (\sup _{t\in [0,T]}\mathcal {W}_2(L^N(\underline{X}_t^{\underline{\hat{\alpha }}}), L^N(\underline{\mathcal {X}}_t)) >\varepsilon \Big ) = - \infty .\) 
It then follows from [8]} that the sequence \(L^N(\underline{X}^{\underline{\hat{\alpha }}})\)  satisfies the LDP with rate function \(\mathcal {I}\) .

Let us now turn to the large deviation principle for the \(N\) -Nash equilibrium \((\hat{\alpha }^{i,N})_{i=1,\dots ,N}\) .
The difficulty here is the fact that \(\hat{\alpha }^{i,N}\)  is not a function of \(X^{i,\underline{\hat{\alpha }}}\)  and \(L^N(\underline{X}^{\underline{\hat{\alpha }}})\)  only, it also depends on the process \(\zeta _t^{i,N}\) , see (REF ).
Nevertheless, using the contraction principle, we will again prove the LDP for an auxiliary sequence that is exponentially equivalent to \(L^N(\underline{\hat{\alpha }})\) .
Define the auxiliary process
\(\alpha ^{i,N}_t := \Lambda \Big (t,\mathcal {X}^{i,N}_t, V\big (t,\mathcal {X}^{i,N}_t, L^N(\underline{\mathcal {X}}_t)\big ),L^N(\underline{ \mathcal {X}}_t) \Big ).\) 
It follows by continuity of \(\Lambda \)  and \(V\) , the convergence of \((\mathcal {X}^{i,N})_{N\ge 1}\)  to \(X\)  and the representation (REF ) of the mean field equilibrium \(\hat{\alpha }\)  that
\(\alpha ^{i,N}_{t} \rightarrow \hat{\alpha }_t \quad \text{in}\quad \mathbb {L}^2.\) 
We will now use the contraction principle to show that \((L^N(\underline{\alpha }_t))_{N\ge 1}\)  satisfies the LDP.
Consider the function \(\Psi \)  mapping \(\mathcal {P}(\mathbb {R}^\ell )\)  to \(\mathcal {P}(\mathbb {R}^\ell )\)  and defined as
\(\Psi ( t,\mu ) := \mu \circ \Lambda \big (t, \cdot , V(t, \cdot , \mu ),\mu \big )^{-1} .\) 
The function \(\Psi \)  is continuous.
In fact, given a sequence \((\mu ^n)_{n\ge 1}\)  converging to \(\mu \)  in the weak topology, and a bounded, Lipschitz-continuous test function \(f\) , we have
\(\int _{\mathbb {R}^\ell }f(x)\Psi (t,\mu ^n)(dx) &= \int _{\mathbb {R}^\ell }f\Big (\Lambda (t, x, V(t, x, \mu ^n),\mu ^n)\Big )\mu ^n(dx).\) 

Since \(\Lambda \)  and \(V\)  are continuous on \(\mathcal {P}(\mathbb {R}^\ell )\) , the functions \(f^n(x):=f\Big (\Lambda (t, x, V(t, x, \mu ^n),\mu ^n)\Big )\)  define a sequence of bounded continuous functions converging to \(f\Big (\Lambda (t, x, V(t, x, \mu ),\mu )\Big )\)  pointwise.
Since \(f^n\)  is uniformly bounded and uniformly Lipschitz, it follows that
\(\int _{\mathbb {R}^\ell }f(x)\Psi (t,\mu ^n)(dx)&= \int _{\mathbb {R}^\ell }f^n(x)\mu ^n(dx) \rightarrow \int _{\mathbb {R}^\ell }f\Big (\Lambda (t, x, V(t, x, \mu ),\mu )\Big )\mu (dx),\) 

showing that \(\Psi \)  is continuous.
Now, by the definition of \(\Psi \) , we have
\(L^N(\underline{\alpha }_t) = \Psi (t, L^N(\underline{\mathcal {X}}_t))\) 
and
similarly, since the mean field equilibrium \(\hat{\alpha }\)  satisfies the representation \(\hat{\alpha }_t =\Lambda (t,X_t, Y_t, \mathcal {L}(X_t)) = \Lambda \big (t, X_t, V(t, X_t, \mathcal {L}(X_t)), \mathcal {L}(X_t) \big )\) , it follows that
\(\mathcal {L}(\alpha _t) = \Psi (t, \mathcal {L}(X_t)).\) 
Using the fact that the sequence \(L^N(\underline{\mathcal {X}}_t)\)  satisfies the LDP with rate function
\(\mathcal {I}_t(\mu ) := \inf _{\theta \in \mathcal {P}(\mathcal {C}^\ell ): \theta _t = \mu }\mathcal {I}(\theta )\) 
and that \(\Psi \)  is continuous, it follows by the contraction principle that \(L^N(\underline{\alpha }_t)\)  satisfies the LDP as well.
In fact, for every bounded continuous function \(F:\mathcal {P}(\mathbb {R}^m) \rightarrow \mathbb {R}\) , the function \(F\circ \Psi (t,\cdot )\)  is again bounded continuous and therefore we have
\(-\frac{1}{N}\log \mathbb {E}\Big [\exp (-NF(L^N(\underline{\alpha }_t)))\Big ] &= -\frac{1}{N}\log \mathbb {E}\Big [\exp \big (-NF\circ \Psi (t,L^N(\underline{\mathcal {X}}_t))\big )\Big ]\\&\rightarrow \inf _{\mu \in \mathcal {P}(\mathbb {R}^\ell )}(F\circ \Psi (t,\mu ) - \mathcal {I}_t(\mu )) \\& = \inf _{\nu \in \mathcal {P}(\mathbb {R}^m)}(F(\nu ) - \widetilde{\mathcal {I}}(\nu ))\) 

with
\(\widetilde{\mathcal {I}}(\nu ) :=\inf _{\theta \in \mathcal {P}(\mathcal {C}^\ell ): \Psi (t,\theta _t) = \nu }\mathcal {I}(\mu ).\) 
Similar to the proof of the LDP for \(L^N(\underline{X}^{\underline{\hat{\alpha }}})\) , it remains to show that the sequences \((L^N(\underline{ \alpha }_t))_N\)  and \((L^N(\underline{\hat{\alpha }}_t))_N\)  are exponentially close and that the rate function \(\mathcal {\widetilde{I}}\)  has compact sublevel sets.
The latter property follows from the fact that for every \(x\ge 0\)  it holds \(\lbrace  \nu \in \mathcal {P}(\mathbb {R}^m): \mathcal {\widetilde{I}}(\nu )\le x\rbrace  = \Psi \big (t,\lbrace  \theta _t: \theta \in \mathcal {P}_2(\mathbb {R}^\ell ),\,\, \mathcal {I}(\theta )\le x\rbrace  \big ) \)  and \(\Psi (t,\cdot )\)  is a continuous function for the weak topology.
Let us now show exponential closeness.
To this end, we first show that the function \(\Psi (t,\cdot )\)  is Lipschitz continuous with respect to the 2-Wasserstein topology.
By Kantorovich's duality theorem, see [9]}, for every \(\xi ,\xi ^{\prime } \in \mathcal {P}_2(\mathbb {R}^\ell )\)  we have
\(\mathcal {W}_2(\Psi (t,\xi ), \Psi (t,\xi ^{\prime })) &= \sup \bigg (\int _{\mathbb {R}^\ell } h_1(x)\Psi (t,\xi )(dx) - \int _{\mathbb {R}^\ell } h_2(x^{\prime })\Psi (t,\xi ^{\prime })(dx^{\prime }) \bigg )\\& = \sup \bigg (\int _{\mathbb {R}^\ell } h_1(\Lambda (t,x,V(t,x,\xi ), \xi ))\xi (dx)\\&\qquad - \int _{\mathbb {R}^\ell } h_2(\Lambda (t, x^{\prime }, V(t,x,\xi ^{\prime }),\xi ^{\prime }))\xi ^{\prime }(dx^{\prime }) \bigg )\) 

with the supremum being taken over the set of bounded continuous functions \(h_1,h_2:\mathbb {R}^\ell \rightarrow \mathbb {R}\)  such that \(h_1(x) -h_2(x^{\prime }) \le |x-x^{\prime }|^2\)  for every \(x, x^{\prime }\in \mathbb {R}^\ell \)  which, by Lipschitz continuity of \(\Lambda \)  and \(V\)  imply that \(h_1\Big (\Lambda (t, x, V(t,x,\xi ), \xi ) \Big ) - h_2\Big ( \Lambda (t, x^{\prime },V(t,x^{\prime },\xi ^{\prime }), \xi ^{\prime }) \Big )\le C|x -x^{\prime }|^2 + C\mathcal {W}_2(\xi ,\xi ^{\prime })\)  for some constant \(C>0\) .
This shows that
\(\mathcal {W}_2(\Psi (t,\xi ), \Psi (t,\xi ^{\prime })) \le C\sup \bigg ( \int _{\mathbb {R}^\ell } \tilde{h}_1(x)\xi (dx) - \int _{\mathbb {R}^\ell } \tilde{h}_2(x^{\prime })\xi ^{\prime }(dx^{\prime }) \bigg ) + C\mathcal {W}_2(\xi ,\xi ^{\prime })\) 

with the supremum over functions \(\tilde{h}_1,\tilde{h}_2\)  such that \(\tilde{h}_1(x) - \tilde{h}_2(x^{\prime }) \le C|x-x^{\prime }|^2\) .
Hence, applying Kantorovich duality once again yields
\(\mathcal {W}_2(\Psi (t,\xi ), \Psi (t,\xi ^{\prime })) \le C\mathcal {W}_2(\xi , \xi ^{\prime }).\) 

Thus, we have
\(\mathbb {P}\Big ( \mathcal {W}_2^2\big (L^N(\underline{\hat{\alpha }}_t), L^N(\underline{\alpha }_t) \big )> \varepsilon \Big )&\le \mathbb {P}\Big (\mathcal {W}_2^2\big (L^N(\underline{X}^{\underline{\alpha }}_t, L^N(\underline{\mathcal {X}}_t)\big ) > \varepsilon \Big ) \\&\le Ce^{-\varepsilon ^2N^2}\) 

for some constant \(C>0\) .
Therefore, it follows that
\(\lim _{N\rightarrow \infty }\frac{1}{N}\log \mathbb {P}\Big (\sup _{t\in [0,T]}\mathcal {W}_2^2(L^N(\underline{\hat{\alpha }}), L^N(\underline{\alpha _t})) >\varepsilon \Big ) = - \infty .\) 
This concludes the proof.
